Local description

its derivative
Df is invertible (as the matrix of all \partial f_i/\partial x_j, 1
\leq i,j \leq n), which means the same as having non-zero Jacobian
determinant.

Remarks:

Condition 2 excludes diffeomorphisms going from dimension n to a different
dimension k (the matrix Df would not be square hence certainly not
invertible).

A differentiable bijection is not necessarily a diffeomorphism,
e.g. f(x)=x^3 is not a diffeomorphism from \mathbb to itself
because its derivative vanishes at 0 (and hence its inverse is not
differentiable at 0).

Now, f from M to N is called a diffeomorphism if
in
coordinates charts it satisfies the definition above. More
precisely, pick any cover of M by compatible
coordinate charts, and do the same for N. Let \phi and \psi be
charts on M and N respectively, with U being the image of \phi and
V the image of \psi. Then the conditions says that the map \psi f
\phi^ from U to V is a diffeomorphism as in the definition above
(whenever it makes sense). One has to check that for every couple
of charts \phi, \psi of two given
atlases, but once checked, it will be true for any other
compatible chart. Again we see that dimensions have to agree.

Diffeomorphism group

The diffeomorphism group of a manifold
is the group of all its automorphisms (diffeomorphisms to itself).
For dimension greater than or equal to one this is a large group.
For a connected
manifold M the diffeomorphisms act
transitively on M: this is true locally because it is true in
Euclidean
space and then a topological argument shows that given any p
and q there is a diffeomorphism taking p to q. That is, all points
of M in effect look the same, intrinsically. The same is true for
finite
configurations of points, so that the diffeomorphism group is k-
fold
multiply transitive for any integer k ≥ 1, provided the
dimension is at least two (it is not true for the case of the
circle or real line).
This group can be given the structure of an infinite dimensional
Lie group, modeled on the space of vector
fields on the manifold. In general, this will not be a Banach
Lie group, and the exponential map will not be a local
diffeomorphism.

The (orientation-preserving) diffeomorphism group
of the circle is pathwise connected. This can be seen by noting
that any such diffeomorphism can be lifted to a diffeomorphism f of
the reals satisfying f(x+1) = f(x) +1; this space is convex and
hence path connected. A smooth eventually constant path to the
identity gives a second more elementary way of extending a
diffeomorphism from the circle to the open unit disc (this is a
special case of the Alexander
trick).

If M is an oriented smooth closed manifold, it
was conjectured by Smale that the identity
component of the group of orientation-preserving
diffeomorphisms is simple. This had first been proved for a product
of circles by Michel
Herman; it was proved in full generality by Thurston.

Homeomorphism and diffeomorphism

It is easy to find a
homeomorphism which is not a diffeomorphism, but it is more
difficult to find a pair of homeomorphic manifolds that
are not diffeomorphic. In dimensions 1, 2, 3, any pair of
homeomorphic smooth manifolds are diffeomorphic. In dimension 4 or
greater, examples of homeomorphic but not diffeomorphic pairs have
been found. The first such example was constructed by John Milnor
in dimension 7, he constructed a smooth 7-dimensional manifold
(called now Milnor's
sphere) which is homeomorphic to the standard 7-sphere but not
diffeomorphic to it. There are in fact 28 oriented diffeomorphism
classes of manifolds homeomorphic to the 7-sphere (each of them is
a fiber
bundle over the 4-sphere with fiber the 3-sphere).

Much more extreme phenomena occur for 4-manifolds: in
the early 1980s, a combination of results due to Simon
Donaldson and Michael
Freedman led to the discovery of exotic R4s:
there are uncountably many pairwise non-diffeomorphic open subsets
of \mathbb^4 each of which is homeomorphic to \mathbb^4, and also
there are uncountably many pairwise non-diffeomorphic
differentiable manifolds homeomorphic to \mathbb^4 which do not
embed smoothly in \mathbb^4.